#7623249

order	512
nilpotency class	4
isoclinism	isoclinic class of #33607
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂²	C₂³×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1408 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(11,19)(12,20)(13,23)(15,25)(17,26)(18,30)(21,35)(22,33)(24,34)(27,36)(28,40)(29,39)(31,32)(37,38);(2,5)(6,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,25)(20,26)(21,28)(22,27)(23,30)(24,29)(31,37)(32,38)(33,36)(34,39)(35,40);(9,11,20,32,13,21,24,33,14,22,34,35,23,31,12,19)(10,15,26,38,18,28,29,36,16,27,39,40,30,37,17,25);(1,3,4,7)(2,6,5,8);(10,16)(15,27)(17,29)(18,30)(25,36)(26,39)(28,37)(38,40)

Polycyclic presentation:

2^9|c^2=gh,d^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,g^a=gh,h^a=hi,e^b=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232015, #72232017, #72232018, …
maximal normal subgroup of	#72232015, #72232017, #72232018, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530445);